PSSA Mathematics Glossary

Transcription

INTRODUCTION
The PSSA Mathematics Glossary includes terms and definitions associated with the Mathematics Assessment Anchors
and Eligible Content aligned to the Pennsylvania Core Standards. The terms and definitions included in the glossary
are intended to assist Pennsylvania educators in better understanding the PSSA Assessment Anchors and Eligible
Content. The glossary does not define all possible terms included on an actual PSSA administration, and it is not
intended to define terms for use in classroom instruction for a particular grade level or course.
This glossary provides definitions for terms in Grades 3–8. In addition to the term and its definition, the grade level at
which the term would first be introduced is included. For terms not specifically found within the Assessment Anchors
and Eligible Content, an asterisk (*) is found next to the grade level, indicating that the grade is an estimated grade for
that term.
Pennsylvania Department of Education
Page 2
Pennsylvania System of School Assessment: Mathematics
Assessment Anchors and Eligible Content Glossary
June 2014
Term
Definition
Grade
Absolute Value
The magnitude of an expression under consideration. The absolute value of a
number is the distance the number is from 0 on the number line. The notation
used to designate the absolute value of expression w is UwU, which is read as
“the absolute value of w.”
For example:
• U{12U = 12
• U451U = 451
6
U U
1 =91
• U9 }
U 2}
2
4
• {4
} =}
7 7
• U{3 + 2U = U{1U = 1
See also Magnitude.
Acute Angle
An angle with a measure greater than 0° and less than 90°.
Acute Triangle
A triangle in which all interior angles are acute angles.
4
4*
Acute Triangle
See also Obtuse Triangle and Right Triangle.
Addend
A number or expression that is added to another number or expression.
For example:
• In the equation 2 + 7 = 9, the 2 and 7 are addends.
• In the equation + 9 = 24, the and 9 are addends.
• In the equation (2 + 3) + 6 = 11, the expression (2 + 3) and the 6 are
addends.
3
An asterisk (*) found next to the grade level indicates that the term is not specifically found within the
Assessment Anchors and Eligible Content.
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Term
Definition
Additive Inverse
An expression that can be added to a given expression so that their sum is
zero.
June 2014
Grade
For example:
• 82 and {82 are additive inverses.
• (19 × 3) and {(19 × 3) are additive inverses.
6*
See also Opposite of a Number.
Adjacent Angles
Two angles with a common side and a common vertex but no overlap.
1
4
2
Adjacent Angles
In the picture, angle 1 and angle 2 are adjacent angles.
Algebraic
Expression
A mathematical expression that contains one or more variables.
For example:
• 7x + 3
6
2w – 17
• }
19r + 7m
• {4xy
See also Numerical Expression.
Alternate Exterior
Angles
Two nonadjacent angles on opposite sides of a transversal and on the exterior
of a pair of parallel lines intersected by the transversal.
1
7
2
Alternate Exterior Angles
In the picture, angle 1 and angle 2 are alternate exterior angles.
See also Alternate Interior Angles and Corresponding Angles.
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June 2014
Term
Definition
Grade
Alternate Interior
Angles
Two nonadjacent angles on opposite sides of a transversal and between a pair
of parallel lines intersected by the transversal.
1
2
7
Alternate Interior Angles
In the picture, angle 1 and angle 2 are alternate interior angles.
See also Alternate Exterior Angles and Corresponding Angles.
Angle
The inclination between intersecting lines, line segments, and/or rays often
measured in degrees (e.g., a 90° inclination is a right angle). The figure is often
represented by two rays that have a common endpoint.
Angles are generally named using three points: one point from each ray, with
the common endpoint in between (e.g., angle ABC consists of ray BA and
ray BC). The symbol for an angle is / and is generally used in conjunction with
the three letters (e.g., angle ABC can also be written as /ABC).
4
A
B
C
Angle ABC (/ABC)
Area
The measure, in square units, of the interior of a plane figure. Units such as
square feet (sq ft) and square centimeters (cm2 ) are used to measure area.
3
Area of a Rectangle
In the picture, each small square represents 1 square unit and the area of the
rectangle is 12 square units.
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June 2014
Term
Definition
Array
A rectangular arrangement of objects, symbols, or numbers. An array may or
may not display vertical or horizontal grid lines. In an array, all rows are the
same length and all columns are the same length.
37
39
38
43
44
39
Associative
Property (Addition
or Multiplication)
Grade
132
152
136
171
175
150
12
24
12
24
41
13
40
43
35
44
45
35
10
4
6
18
4
10
Array
0
0
1
1
0
0
2
2
1
3
11
4
27
13
13
29
25
30
14
23
16
19
30
9
10
32
49
45
39
49
3
The property that asserts the grouping of adjacent addends or factors is
irrelevant. That is, (a + b) + c = a + (b + c) and a × (b × c) = (a × b) × c.
For example:
• by the associative property of addition: (3 + 9) + 2 = 3 + (9 + 2)
• by the associative property of multiplication: (3 × 9) × 2 = 3 × (9 × 2)
3
Note: by contrast, subtraction and division do not hold true under the
associative property
See also Commutative Property (Addition or Multiplication).
Average
See Mean.
3*
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June 2014
Term
Definition
Grade
Axis
A vertical or horizontal number line, both of which are used to define a
coordinate grid. The horizontal axis is the x-axis, and the vertical axis is the
y-axis. The plural of axis is axes.
The intersection of the two axes occurs at 0 of both number lines. This
intersection is the origin, which is designated by the ordered pair (0, 0).
The axes divide the plane into four quadrants.
When a point on a coordinate grid is named with an ordered pair, such as
(5, 11), the first number (5) is the x-coordinate and the second number (11) is
the y-coordinate.
When representing an equation or other relation, the input values are on the
x-axis and the output values are on the y-axis.
y
6
5
4
5
3
2
1
–6 –5 –4 –3 –2 –1
–1
1
2
3
4
5
6
x
–2
–3
–4
–5
–6
x-Axis and y-Axis
See also Origin, Quadrant, Ordered Pair, Independent Variable, and
Dependent Variable.
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June 2014
Term
Definition
Grade
Bar Graph
A type of data display that represents a frequency distribution. The class
intervals (buckets) in a bar graph represent categorical data. Bar graphs may
either be vertical or horizontal.
The class intervals in a vertical bar graph are located on the x-axis and form
the bases of nonadjacent rectangular bars. Frequencies are listed on the
y-axis.
The class intervals in a horizontal bar graph are located on the y-axis and
form the bases of nonadjacent rectangular bars. Frequencies are listed on
the x-axis.
The class interval representation of categorical data rather than numerical
data, and nonadjacent bars rather than contiguous bars, are distinguishing
features of a bar graph in contrast to a histogram.
3
Carnival Prizes
Number Won
4
3
2
1
0
balloon
lollipop
pencil
ring
Types of Prizes
Bar Graph
Bivariate Data
Data or observations represented by two variables. The variables may or may
not be independent.
For example:
• Age of players on a team and gender of the players (independent
bivariate variables)
• Gallons of gasoline purchased and cost of the gasoline purchased
(dependent bivariate variables)
8
Page 8
June 2014
Term
Definition
Grade
Box-and-Whisker
Plot
A plot that visually represents a set of data. A rectangle (the box) is used to
represent the dispersion of points between the first and third quartiles, and
line segments (the whiskers) are used to represent the dispersion of points
between the minimum value and the first quartile and between the maximum
value and the third quartile. A line segment drawn within the box represents
the median value.
The plot provides a five-number summary of the data—the minimum, first
quartile, median, third quartile, and maximum values. This five-number
summary of the data is specified on the plot or is evident from a number line
drawn above or below the plot.
The example below shows a horizontal box-and-whisker plot. Box-andwhisker plots can also be vertically oriented.
6
median
minimum
0
2
4
6
ﬁrst
quartile
third
quartile maximum
8 10 12 14 16 18 20 22 24 26 28 30 32
Box-and-Whisker Plot
See also Median, Quartile, and Interquartile Range.
Chance Event
(Random Event)
An event that leads to an outcome that cannot be determined prior to
completion of the event but can be described probabilistically without an
apparent cause (i.e., a probability of an outcome can be assigned).
For example: flipping a coin is a chance/random event. The outcome cannot
7
be determined prior to flipping the coin, but all possible outcomes can be
1
assigned probability values (i.e., P(Head) = 1
}, P(Tail) = } ). Other examples of
2
2
chance/random events include rolling a number cube or a “blind” drawing.
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June 2014
Term
Definition
Grade
Chord
A line segment with endpoints on a circle. If a chord contains the center of the
circle, it is referred to as a diameter of the circle.
chord AB
8*
A
B
Chord
Circle
A two-dimensional (plane) figure for which all points are the same distance
from its center. Informally, a perfectly round shape. A circle is identified by its
center point.
7
C
Circle C
Circumference
The distance around a circle. The circumference of a circle is analogous to the
perimeter of a polygon.
Coefficient
The constant by which a variable is multiplied.
For example:
• In the expression 6x, 6 is the coefficient.
• In the expression 27ab, 27 is the coefficient.
Combination
7
6
A unique set or group of objects, symbols, numbers, etc. Only the contents of
the set, not the order or arrangement, determine a combination.
For example:
• Contents of the sets {a, 5, cat} and {5, a, cat} represent the same
combination. Placing these elements in a different order does not create
a new combination.
• Contents of the sets {w, 12, dog} and {w, 23, fish} are different
combinations. Because one or more elements are different, the
combinations are different.
6*
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June 2014
Term
Definition
Grade
Commutative
Property (Addition
or Multiplication)
The property that asserts the order of adding adjacent addends or multiplying
adjacent factors is irrelevant. That is, a + b = b + a and a × b = b × a.
For example:
• by the commutative property of addition: 7 + 4 = 4 + 7
• by the commutative property of multiplication: 7 × 4 = 4 × 7
3
Note: by contrast, subtraction and division do not hold true under the
commutative property
See also Associative Property (Addition or Multiplication).
Complementary
Angles
Two angles for which the sum of their measures is 90°.
If two complementary angles are also adjacent angles, they form a right angle.
Each of two complementary angles is referred to as the complement of the
other angle (e.g., a 65° angle is the complement of a 25° angle).
7
25º
65º
Complementary Angles
See also Supplementary Angles.
Complex Fraction
A fraction in which the numerator, denominator, or both are also fractions.
For example:
3
}
7
• }
24
9
}
11
• }
5
}
7
7
37
• }
13
}
29
Composite Number A whole number greater than 1 that is not a prime number.
4
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Term
Definition
Compound Event
An event composed of two or more contributing events.
June 2014
Grade
For example:
• Occurrence of rain on a Saturday—contributing events are (1) it rains
and (2) it is Saturday.
• A tossed coin results in two heads—contributing events are (1) head on
the first toss and (2) head on the second toss.
7
A compound event is often a consideration in determining probability
(compound probability).
Cone
A three-dimensional (solid) figure that has a circular base and one vertex. A
cone has two faces: the circular base and the lateral face.
vertex
8
cone
(On the PSSA, it may be assumed all cones are right cones unless otherwise
specified.)
Congruent
Geometric figures that have the same size and the same shape. Congruent
figures may have different orientations.
• Congruent angles have the same degree measure.
• Congruent segments are the same length.
In the case of congruent polygons, the identifying vertices of the two polygons
refer to corresponding angles.
8
Congruent Polygons
Constant of
Proportionality
The constant multiplier by which one variable in a proportional relationship
is related to the other variable. Constant of proportionality and unit rate are
equivalent.
7
For example: if an airplane travels at a constant rate of 250 miles per hour, the
constant of proportionality in the relation of distance (d) to time (t) is 250
(i.e., d = 250t).
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June 2014
Term
Definition
Grade
Coordinate Grid
(Coordinate Plane)
A plane that has been divided into spaces determined by perpendicular
number lines in the plane. The perpendicular number lines represent the axes
of the coordinate grid. The intersection of the perpendicular number lines
is the origin and is used to determine points named with ordered pairs of
numbers.
y
6
5
4
3
2
1
–6 –5 –4 –3 –2 –1
–1
–2
–3
–4
–5
–6
1 2 3 4 5 6
x
5
Coordinate Grid (or Coordinate Plane)
In this picture the axes are labeled as x and y.
The phrases coordinate grid and coordinate plane are interchangeable.
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June 2014
Term
Definition
Grade
Correlation
A measure of the correspondence between the change in one element in a
bivariate data set and the change in the related element in the bivariate data
set.
Positive Correlation: when the increase in value of one element of a bivariate
data set corresponds to an increase in value of the related element in the data
set. For example, if an increase in the outdoor temperature corresponds to an
increase in the number of ice-cream cones sold, the correlation is positive.
Negative Correlation: when the increase in value of one element of a bivariate
data set corresponds to a decrease in value of the related element in the data
set. For example, if an increase in the outdoor temperature corresponds to a
decrease in the number of ice skates sold, the correlation is negative.
Correlation Coefficient: a number (r), such that –1 ≤ r ≤ 1, that provides a
measure of the degree of correlation between elements in bivariate data sets.
For perfect negative correlation, r = –1; for no correlation, r = 0; and for perfect
positive correlation, r = 1. Correlation coefficient (r) can also be conceptualized
as a numerical measure, such that –1 ≤ r ≤ 1, of the correlation between points
in a data set and the related points predicted by a line of best fit.
–1 < r < 0
r = –1
0 < r < +1
8
r = +1
r=0
Correlations
See also Line of Best Fit.
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June 2014
Term
Definition
Grade
Corresponding
Angles
Pairs of angles having the same relative position in geometric figures.
L
V
Example 1:
9
U
15
12
W
20
K
M
D
A
Example 2:
B
E
C
F
Corresponding Angles
In Example 1, /UVW and /KLM are corresponding angles. In Example 2,
/ABC and /DEF are corresponding angles.
7
Corresponding angles can also refer to two nonadjacent angles on the same
side of a transversal with one angle between the lines cut by the transversal
(interior) and one angle outside the lines cut by the transversal (exterior).
Corresponding angles are in the same relative position with respect to the
intersections of a transversal and two parallel lines.
1
2
Corresponding Angles (parallel lines cut by a transversal)
In the picture, angle 1 and angle 2 are corresponding angles.
For reference related to the second definition, see Alternate Exterior Angles
and Alternate Interior Angles.
Counting Number
Any number from the set of numbers represented by {1, 2, 3, …}. A counting
number is sometimes referred to as a “natural number” or a “positive integer.”
6*
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Term
Definition
Cube
A rectangular solid with exactly six congruent square faces.
June 2014
Grade
7
Cube
Cube Root
One of three equal factors (roots) of a number or expression. The cube root of
a number or expression has the same sign (positive/negative) as the number
under the radical. Informally, it can be thought of as “the number that, when
multiplied by itself, and then multiplied by itself again, has a product equal to a
given number.”
For example:
8
3}
• Ï 8 = 2 since 2 × 2 × 2 = 8
3}
• Ï {64 = {4 since {4 × {4 × {4 = {64
3}
• Ï 0.343 = 0.7 since 0.7 × 0.7 × 0.7 = 0.343
3}
• Ï 125w6 = 5w2 since 5w2 × 5w2 × 5w2 = 125w6
Cylinder
A three-dimensional figure with two circular bases that are parallel and
congruent. A cylinder has three faces: the two circular bases and the lateral
face.
8
Cylinder
(On the PSSA, it may be assumed all cylinders are right cylinders unless
otherwise specified.)
Page 16
Term
Definition
Decagon
A polygon with exactly 10 sides.
June 2014
Grade
4*
Decagons
Decimal Notation
A number written with base 10 place values that are smaller than one
(e.g., tenths, hundredths). These place values are written to the right of a
decimal point (e.g., 0.91, 25.624).
Decimal notation is different from fraction notation. For example:
• decimal notation: 0.25
4
• fraction notation: 1
}
4
Degree (angle)
A unit of measure for the inclination of an angle. It is represented by the
symbol ° and is used in conjunction with the number (e.g., 30° is read as
1 of the angle inclination change
“thirty degrees”). Each degree represents }
360
from two rays being on top of one another (0°) to a complete revolution (360°)
4
about the shared endpoint of the two rays.
Degree
(temperature)
A unit of measure for temperature. It is represented by the symbol ° and is
used in conjunction with the number (e.g., {40° is read as either “negative forty
degrees” or “forty degrees below zero”).
1 of the
In the Fahrenheit (F) temperature scale, each degree represents }
180
temperature change between the freezing point of water (32°F) and the boiling
point of water (212°F).
6
1 of the
In the Celsius (C) temperature scale, each degree represents }
100
temperature change between the freezing point of water (0°C) and the boiling
point of water (100°C). A less common name for this temperature scale is
centigrade. Celsius is the officially recognized PSSA term.
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Term
Definition
Denominator
The divisor in a ratio or fraction.
June 2014
For example: in the fraction 7
}, 9 is the denominator.
9
Often students first learn the informal definition of denominator as “the bottom
number” in a ratio or fraction.
Grade
3
See also Numerator.
Dependent Events
Two or more events in which the outcome of one event affects or influences
the outcome of the other event(s). Sometimes, these events can happen at the
same time.
For example:
• Event 1: Picking a card from a deck; Event 2: Picking a second card
from the same deck without replacing the first card.
• Event: Selecting two colored markers at the same time from a set of
markers.
7*
(On the PSSA, it may be assumed that events occurring at the same time
is the same as events occurring one at a time without replacement unless
See also Independent Events.
Dependent Variable The variable in a relation that represents a value determined by the
independent variable. The dependent variable is sometimes referred to as
the “output variable.” When a relation is written as a set of ordered pairs, the
y-coordinate corresponds to the dependent variable.
6
For example: In the relation y = 3x – 8, when the independent variable (x) is
replaced by the number 5, the value of the dependent variable (y) is 7.
See also Independent Variable.
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June 2014
Term
Definition
Grade
Diameter
A line segment that has endpoints on a circle and passes through the center
of the circle. A diameter is a chord that contains the center of the circle.
Diameter
5*
In common usage, diameter occasionally refers not only to the line segment
but also to the length of the line segment that constitutes the diameter.
(On the PSSA, it may be assumed that diameter is the line segment, not the
measurement of the line segment unless otherwise specified. If there is a
context in which diameter is intended to imply a measurement, the context
must clearly, absolutely, and indisputably make that assertion.)
See also Radius.
Difference
The result when one number is subtracted by another number (i.e., the
“answer” to a subtraction computation). Unless otherwise specified, it may be
assumed that the difference is the absolute value of the subtraction (e.g., the
difference of 3 and 7 and the difference of 7 and 3 are both 4).
7
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June 2014
Term
Definition
Grade
Dilation
A nonrigid transformation in which linear measurements may change but the
proportional relationships of those measurements are preserved (i.e., length
measurements in the dilated image remain uniformly proportional to length
measurements in the original figure).
In a dilation, all the lengths of a figure are multiplied by a common scale factor.
Angle measurements in a dilation do not change.
Scale Factor of 2
Original Figure
Dilated Figure
A'
A
Dilation
All dilations have a point of emanation, or center of dilation. When a shape
on a coordinate grid is dilated, the scale factor is applied to the difference
between the vertices of the figure and the point of emanation.
y
5
4
C'
A'
C
8
Dilated Figure
B'
3
2
B
Original
Figure
A1
–5 –4 –3 –2 –1
–1
1
2
3
4
5
x
–2
–3
–4
–5
Dilation (on a coordinate grid)
(On the PSSA, it may be assumed the point of emanation on a coordinate grid
is the origin unless otherwise specified.)
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June 2014
Term
Definition
Grade
Distributive
Property
When a single-term expression is being multiplied by a sum or difference, the
single-term expression can be multiplied by each term before finding the sum
or difference. That is, a(b + c) = ab + ac or a(b – c) = ab – ac.
6
For example:
• 5(7 + 4) = 5(7) + 5(4) = 35 + 20
• w(9 – 3) = 9w – 3w
Dividend
When dividing one number by another number, the number that is being
divided.
For example, in the expression 24 ÷ 6, the number 24 is the dividend.
3
See also Divisor.
Divisor
When dividing one number by another number, the number by which another
number is divided.
For example, in the expression 24 ÷ 6, the number 6 is the divisor.
4
See also Dividend.
Edge
The line segment formed by the intersection of two faces of a threedimensional (solid) figure. For example, a cube has 12 edges.
edges
5
Edges of a Cube
In the picture, each solid line segment and each dashed line segment
represents an edge of the cube.
Equation
A mathematical sentence or statement relating two equal expressions.
When written in mathematical notation, an equation always contains an
equal sign (=).
Examples of equations:
• 4 + 15 = 19
• w + 13 = 17 × 12
3
27
• –14
13
(On the PSSA, an equation may be written either horizontally or vertically.)
Page 21
June 2014
Term
Definition
Grade
Equilateral Triangle
A triangle where all sides are the same length (i.e., the sides are congruent).
Each of the angles in an equilateral triangle is 60°. Thus, the triangle is also
equiangular.
A
4*
B
C
Equilateral Triangle
A less common name for this triangle is equiangular, since all the angles are
equal in measure. Equilateral is the officially recognized PSSA term.
See also Isosceles Triangle and Scalene Triangle.
Equivalent
Two or more mathematical statements, expressions, or other representations
that have the same value.
Equivalent mathematical statements, expressions, or other representations,
including geometric figures, are interchangeable in the setting in which they
exist.
3
For example:
• The expressions 2 + 9 and 2 + 3 × 3 are equivalent expressions.
• The sequences 4, 8, 12, 16, … and 2 × 2, 2 × 4, 2 × 6, 2 × 8, … are
equivalent sequences.
• Geometric figures are equivalent if they are congruent.
Expanded Form
(Expanded
Notation)
A whole or decimal number written as the sum of single-digit multiples of
powers of 10.
For example:
• 735.2 = 700 + 30 + 5 + 0.2
• 735.2 = 7 × 100 + 3 × 10 + 5 × 1 + 2 × 0.1
(
1
or, 735.2 = 7 × 100 + 3 × 10 + 5 × 1 + 2 × }
10
4
)
• 735.2 = 7 × 102 + 3 × 101 + 5 × 100 + 2 × 10{1
The phrase expanded notation is equivalent to and interchangeable with
expanded form.
Page 22
June 2014
Term
Definition
Experimental
Probability
A likelihood of an outcome based on the number of favorable outcomes that
have occurred compared to the total number of outcomes that have occurred.
An experimental probability is based on a series of trials.
For example, if a coin lands heads on 17 of the 20 times it is flipped, the
Grade
8*
experimental probability of this coin landing heads is 17
}.
20
See also Theoretical Probability.
Expression
A number or variable, the power of a number or variable, or the sum,
difference, product, or quotient of a combination of numbers, variables, and/or
powers of a number or variable.
Examples of expressions:
• 15xy
• 23 × 6 + 51
• 3r}
– 28
• Ï 38
6
Expressions do not contain relations such as =, >, <, etc. Expressions are the
elements that form mathematical sentences, equations, or inequalities, but
they are not mathematical sentences, equations, or inequalities.
Face
Fact Family
A two-dimensional (plane) figure that is one side of a three-dimensional (solid)
figure. The faces make up the surface of the three-dimensional (solid) figure.
For example, the six squares that form a cube are the faces of the cube.
A set of related addition and subtraction equations or related multiplication
and division equations using the same numbers.
For example:
• 9 + 6 = 15, 6 + 9 = 15, 15 – 9 = 6, 15 – 6 = 9
• 3 × 4 = 12, 4 × 3 = 12, 12 ÷ 3 = 4, 12 ÷ 4 = 3
Factor
A whole number that can divide another whole number with no remainder. For
example, 1, 3, 5, and 15 are factors of 15.
Factor Pair
A pair of whole numbers with a product equal to the number under
consideration. For example, the numbers 2 and 7 are a factor pair of 14 since
2 × 7 = 14.
a,
A ratio of two values, numbers, or expressions. It is written in the form }
b
where b is not equal to 0.
Fraction
Function
3*
A relation in which each input value (independent variable) is associated with
exactly one output value (dependent variable).
1*
3
4
3
4
Page 23
Term
Definition
Greatest Common
Factor (GCF)
The greatest factor that two or more numbers have in common.
Heptagon
June 2014
Grade
For example:
• The greatest common factor of 4 and 10 is 2.
• The greatest common factor of 12, 30, and 42 is 6.
• The greatest common factor of 8 and 15 is 1.
6
4*
Heptagons
Less common names for this polygon are septagon and septilateral. Heptagon
is the officially recognized PSSA term.
Hexagon
4*
Hexagons
Page 24
June 2014
Term
Definition
Grade
Histogram
A type of data display that represents a frequency distribution. The class
intervals (buckets) represent numerical data. The class intervals are located on
the x-axis and form the bases of contiguous rectangular bars. Frequencies are
listed on the y-axis.
The class interval representation of numerical data rather than categorical
data, and contiguous bars rather than nonintersecting bars, are distinguishing
features of a histogram in contrast to a bar graph.
Maximum Lake Depth
Number of Lakes
60
50
6
40
30
20
10
0
1–5
6–10 11–15 16–20 21–25
Depth (meters)
Histogram
Hypotenuse
The side opposite the 90° (right) angle in a right triangle. The hypotenuse is
also the longest side in a right triangle.
A
hypotenuse
leg
B
leg
8*
C
Hypotenuse of a Right Triangle
See also Leg (of a Right Triangle).
Page 25
Term
June 2014
Definition
Grade
Identity Property of The property that asserts the product of an original factor times one is equal to
Multiplication
the original factor.
In less formal mathematical phrasing, it is the property that states whenever a
number/variable/expression is multiplied by one, the product is identical to the
original number/variable/expression.
7*
For example:
• 93 × 1 = 93
• 1(3w + 9) = (3w + 9)
Identity Property of The property that asserts the sum of an original addend plus zero is equal to
the original addend.
Addition
In less formal mathematical phrasing, it is the property that states whenever
a number/variable/expression is added to zero, the sum is identical to the
original number/variable/expression.
7*
For example:
• 93 + 0 = 93
• (3w + 9) + 0 = (3w + 9)
Independent
Events
Two or more events, in which the outcome of one event does not influence or
affect the outcome of the other event(s).
For example:
• Event 1: Flipping a coin; Event 2: Picking a card from a deck
• Event 1: Selecting a colored marker; Event 2: Walking to school
7
See also Dependent Events.
Independent
Variable
The variable that is used to determine the value of a relation. The independent
variable is often referred to as the “input variable” of a relation. When a
relation is written as a set of ordered pairs, the x-coordinate corresponds to
the independent variable. The set of values that can be used to replace the
independent variable is named the “domain” of the relation.
For example: To determine the value of the relation y = 3x – 8, input values
replace the independent variable (x).
6
The values of the equation that result from substituting numbers for the
independent variable are often referred to as dependent or output values.
See also Dependent Variable.
Page 26
June 2014
Term
Definition
Inequality
A mathematical sentence that contains an inequality symbol (i.e., >, <, ≥, ≤,
or ≠). It compares two quantities. The symbol > represents greater than, the
symbol < represents less than, the symbol ≥ represents greater than or equal
to, the symbol ≤ represents less than or equal to, and the symbol ≠ represents
not equal to (the symbol ≠ is often used to express which values are not
available to be used for a particular expression or equation).
Grade
6
For example:
• 3+4>6
• 7 + 2 < 11 –
Integer
A counting number, the additive inverse of a counting number, or zero. Any
number from the set of numbers represented by {…, {3, {2, {1, 0, 1, 2, 3, …}.
4*
Interquartile Range The difference between the third quartile and the first quartile in an ordered set
of numerical data. It represents the spread of the middle 50% of a set of data.
For example: For the data set {2, 5, 7, 12, 17, 22, 23}, the first quartile value is
5 and the third quartile value is 22; so, the interquartile range is 22 – 5 = 17.
6
See also Quartile.
Irrational Number
A number that cannot be precisely represented as a fraction written with
integers.
In relation to other real numbers, an irrational number is any real number that
is not a rational number.
When an irrational number is written in decimal notation, the numeral has an
infinite number of non-repeating digits or non-repeating sequence of digits to
the right of the decimal point.
8
For example:
}
• Ï2
• π
• e (base of the natural logarithm)
See also Rational Number.
Page 27
June 2014
Term
Definition
Grade
Irregular Polygon
A polygon that does not have all congruent sides and all congruent angles.
An irregular polygon may have congruent sides and/or congruent angles.
Essentially, an irregular polygon is any polygon that is not a regular polygon.
6
Irregular Polygons
Two special types of irregular polygons are scalene triangles and trapezoids.
Isosceles Triangle
A triangle with two congruent sides, which are called the legs of the isosceles
triangle. The angles opposite the two legs (called base angles) are also
congruent.
A
3 cm
3 cm
4*
B
2 cm
C
Isosceles Triangle
An equilateral triangle is a special type of isosceles triangle.
See also Equilateral Triangle and Scalene Triangle.
The least common multiple of the denominators of two or more fractions.
Least Common
Denominator (LCD)
For example:
3
• The least common denominator of 1
} and } is 20.
4
10
3 , 5 , and 19 is 60.
• The least common denominator of }
}
}
10 12
30
4*
3
• The least common denominator of 3
} and } is 84.
21
4
Page 28
June 2014
Term
Definition
Least Common
Multiple (LCM)
The least whole number that is a common multiple of two or more numbers.
Leg (of an
Isosceles Triangle)
Grade
For example:
• The least common multiple of 4 and 10 is 20.
• The least common multiple of 10, 12, and 30 is 60.
• The least common multiple of 4 and 21 is 84.
6
Each of the two congruent sides of an isosceles triangle. In an equilateral
triangle, any pair of sides may be considered the legs of the triangle.
A
leg
B
4*
leg
C
base
Legs of an Isosceles Triangle
Leg (of a Right
Triangle)
Each of the two sides that form the right angle in a right triangle.
A
leg
hypotenuse
4*
B
leg
C
Legs of a Right Triangle
See also Hypotenuse.
Line
An infinitely long, straight set of points. Informally, it can be thought of as a
path extending in opposite directions with no endpoints. A line is identified by
any two unique points on the line.
A
B
3
Line AB (@###$
AB)
See also Line Segment.
Page 29
June 2014
Term
Definition
Grade
Line of Best Fit
A line drawn on a scatter plot to best estimate the relationship between two
sets of data. It describes the trend of the data. Different measures are possible
to describe the line of best fit. The most common is a line that minimizes the
sum of the squares of the errors (vertical distances) from the data points to
the line.
y
8
x
Line of Best Fit
Line of Symmetry
A line that divides a figure into two parts that are congruent mirror images of
each other.
4
Line of Symmetry
See also Reflection.
Page 30
June 2014
Term
Definition
Grade
Line Plot
A frequency distribution plot in which the data are single points on a number
line and the frequencies are represented by dots, ×’s, or similar notation. The
data may be categorical or numerical. Unless otherwise specified, it may be
assumed that each mark (dot, ×, or similar notation) represents a value of 1.
Baseball Players
×
×
×
×
×
×
×
×
×
×
×
0
1
2
3
×
×
3
4
×
×
5
6
Home Runs
Line Plot
Line Segment
A portion or subset of a line bounded by two endpoints. Informally, a line
segment can be conceptualized as two points on a line and all the points
between them. A line segment is not a line. A line segment is identified by its
endpoints.
A
B
4
}
Line Segment AB (AB)
See also Line.
Linear Relationship A mathematical relationship between two variables that can be represented by
a linear equation (e.g., Ax + By = C).
If points represent a linear relationship, the graph of those points is a straight
line. If a graph is a straight line, the points on the line represent a linear
relationship.
For example:
• Distance traveled (d) in 6 hours at a constant speed (s):
d + {6s = 0 or d = 6s
• Gallons of water (w) in a container (starting volume = 3 gallons) when
g gallons of water are added per hour for 22 hours:
w + {22g = 3 or w = 22g + 3
8
Page 31
June 2014
Term
Definition
Magnitude
A scalar (no units assigned) associated with a quantity. Magnitude is always a
positive number. In general, magnitude is found by determining the absolute
value of the numerical portion of a quantity.
For example:
• The magnitude of 1,200 feet above sea level is 1,200.
Grade
6
1 pounds is 18 1.
• The magnitude of 18 }
}
2
2
• The magnitude of the number 34.931 is 34.931.
See also Absolute Value.
Maximum
The greatest number in a set of data.
For example:
• For the data set {5, 7, 12, 23, 29}, the maximum is 29.
• For the data set {1, 7, 9, 11}, the maximum is 11.
6*
See also Minimum.
Mean
A number found by dividing the sum of a set of numbers by the number of
addends. The terms mean and average are equivalent.
For example:
• For the data set {1, 7, 9, 11}, the sum of the 4 data points is
1 + 7 + 9 + 11 = 28, so the mean is 28 ÷ 4 = 7.
• For the data set {5, 7, 12, 23, 29}, the sum of the 5 data points is
5 + 7 + 12 + 23 + 29 = 76, so the mean is 76 ÷ 5 = 15.2.
6
See also Median and Mode.
Mean Absolute
Deviation
The average of the differences between each data point in a data set and the
mean.
For example:
• For the data set {1, 7, 9, 11}, the mean of the set is 7 and the sum of the
differences between each data point and 7 is 6 + 0 + 2 + 4 = 12; so, the
mean absolute deviation is 12 ÷ 4 = 3.
• For the data set {5, 7, 12, 23, 29}, the mean of the set is 15.2 and the
sum of the differences between each data point and 15.2 is
10.2 + 8.2 + 3.2 + 7.8 + 13.8 = 43.2; so, the mean absolute deviation is
43.2 ÷ 5 = 8.64.
6
Page 32
June 2014
Term
Definition
Measures of
Center
Statistical measures that are intended to provide numerical representations of
the center of a set of numerical data.
Grade
The phrases measure of center and measure of central tendency are
interchangeable.
For example:
• Mean
• Median
• Mode
• Midrange
6
See also Measures of Variability.
Measures of
Variability
Statistical measures that are intended to provide numerical representations of
the variability of a set of numerical data.
For example:
• Range
• Interquartile Range
• Mean Absolute Deviation
• Standard Deviation
6
See also Measures of Center.
Median
The middle number in a set of data ordered from least to greatest (or from
greatest to least). If the data set consists of an even number of entries, the
median is the mean of the two middle entries in the list.
For example:
• For the data set {5, 7, 12, 23, 29}, the middle number of the ordered set
is 12, so the median is 12.
• For the data set {1, 7, 9, 11}, the middle numbers are 7 and 9, so the
median is the mean of 7 and 9, which is 8.
6
See also Mean and Mode.
Minimum
The least number in a set of data.
For example:
• For the data set {5, 7, 12, 23, 29}, the minimum is 5.
• For the data set {1, 7, 9, 11}, the maximum is 11, minimum is 1.
6*
See also Maximum.
Page 33
June 2014
Term
Definition
Mode
The number that occurs most often in a set of data. A set of data may have
more than one mode, or it may have no mode.
For example:
• For the data set {5, 7, 12, 12, 29}, 12 appears the most often, so the
mode is 12.
• For the data set {1, 1, 6, 6, 6, 11, 11, 11, 13, 17}, both 6 and 11 appear
the most often, so the modes are 6 and 11 (note that 1 is not a mode
since it only appears twice).
• For the data set {1, 7, 9, 11}, no number appears more than once, so the
data set does not have a mode.
Grade
6
See also Mean and Median.
Multiple
A number that is divisible by another number with no remainder.
For example:
• 3, 6, 9, 12, and 15 are all multiples of 3
• 1.75, 3.5, 5.25, 7, and 8.25 are all multiples of 1.75
4
Multiples of a number can be found by multiplying the given number by
whole numbers.
Mutually Exclusive
Events
Events that preclude each other. Mutually exclusive events cannot occur
simultaneously.
Mutually exclusive events are always dependent events.
Negative Number
For example:
• Flipping a coin one time and getting heads and tails. The coin landing
heads meant that it could not also have landed tails and vice versa.
• Arriving 10 minutes early and arriving 10 minutes late. If you arrived
10 minutes early, you did not arrive 10 minutes late and vice versa.
7*
The opposite of a positive number (i.e., any number less than 0).
6
Page 34
June 2014
Term
Definition
Grade
Net
A two-dimensional shape or figure that can be folded to form a
three-dimensional (solid) shape or object. It is usually the case that the
fold lines are marked on the net.
The total area of the net is equal to the total surface area of the associated
three-dimensional (solid) shape or object.
6
solid
net
Net
Nonagon
4*
Nonagons
Number Line
A graph that represents the real numbers as ordered points on a line. A
number line may be either horizontal (left and right) or vertical (up and down).
Starting at zero, the positive numbers progress to the right (or up) and the
negative numbers progress to the left (or down).
3
−8 −7 −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 7 8
Number lines serve as the bases of line plots and box-and-whisker plots. In a
coordinate grid, a horizontal number line is used for the x-axis and a vertical
number line is used for the y-axis.
Number Sentence
A mathematical statement that is either an equation or an inequality. A number
sentence is composed of expressions, but it is not an expression. When
written, a number sentence always contains a relation symbol (e.g., =, ≤, >).
3
Page 35
Term
Definition
Numerator
The dividend in a ratio or fraction.
June 2014
Grade
For example: in the fraction 7
}, 7 is the numerator.
9
Often students first learn the informal definition of numerator as “the top
number” in a ratio or fraction.
3
See also Denominator.
Numerical
Expression
A mathematical expression that does not contain a variable.
For example:
• 679 – 12(45)
• 73
5
See also Algebraic Expression.
Obtuse Angle
An angle with a measure greater than 90º and less than 180°.
Obtuse Triangle
A triangle in which an interior angle is an obtuse angle.
4
5*
Obtuse Triangle
See also Acute Triangle and Right Triangle.
Octagon
4*
Octagons
Opposite
(of a Number)
The additive inverse of a number.
For example:
• The opposite of 458 is {458.
6
3 is 3.
• The opposite of { }
}
7 7
Order of
Operations
}
The rules that specify the order in which operations (e.g., +, –, ×, ÷, Ï ) are
performed when more than one operation in a numerical expression or an
algebraic expression is required.
3
Page 36
June 2014
Term
Definition
Grade
Ordered Pair
A pair of numbers or other elements in which the order of recording is
consequential (i.e., order makes a difference). Ordered pairs can be used
to locate points on a coordinate grid. Ordered pairs, both numerical and
non-numerical, are written within a set of parentheses or in an x-y table.
For example:
• (5, 9)
• (insect, ant)
• (x + 2, 3 × w)
•
x
–1
0
1
5
y
0
2
4
On a coordinate grid, the ordered pair (a, b) refers to a point at the intersection
of the vertical line through a on the x-axis and the horizontal line through b on
the y-axis.
Origin
The intersection of the perpendicular axes of a coordinate grid. The origin is
designated by the ordered pair (0, 0).
y
6
5
4
3
2
1
–6 –5 –4 –3 –2 –1
–1
–2
–3
–4
–5
–6
Origin
5
1 2 3 4 5 6
x
Origin
Page 37
June 2014
Term
Definition
Grade
Outlier
A value that is noticeably greater than or less than the observed or expected
values in the data set (i.e., a value that, in the judgment of an observer, is
excessively aberrant).
Unless otherwise specified, it may be assumed that an outlier is a value in a
data set that is 1.5 times the interquartile range less than the first quartile or
1.5 times the interquartile range greater than the third quartile.
For example: the data set {5, 18, 22, 23, 24, 26, 27, 27, 28, 33, 70} has
two outliers:
• 1st quartile = 22, 3rd quartile = 28
• Interquartile range = (28 – 22) = 6
• 1.5 times the interquartile range = 1.5(6) = 9
• 70 – 28 > 9 and 22 – 5 > 9; therefore, both 70 and 5 are outliers.
Visually, an outlier can be seen in a box-and-whisker plot when a whisker
is at least 1.5 times as long as the box representing the data between the
1st quartile and the 3rd quartile. It should be noted that only the minimum
value and/or the maximum value can be identified as outliers using this
method.
8
For example, in the box-and-whisker plot shown below, the box representing
the data between the 1st quartile and the 3rd quartile has a length of 6, the
whisker representing the lower quartile of the data has a length of 17, and the
whisker representing the upper quartile of the data has a length of 42.
0
Parallel Lines
5 10 15 20 25 30 35 40 45 50 55 60 65 70 75
Two or more lines that lie in the same plane and do not intersect.
Parallel Lines
4
For illustration purposes, railroad tracks are often used to represent parallel
lines, whereas a bridge and river under the bridge are often used to represent
skew lines (they do not intersect, but they are not parallel).
Parallelogram
A quadrilateral in which opposite sides are parallel and congruent.
5
Parallelogram
Page 38
Term
Definition
Pentagon
June 2014
Grade
3
Pentagons
Perfect Cube
A number for which the cube root is an integer.
For example:
3}
• Since Ï 27 = 3, 27 is a perfect cube.
3}
• Since Ï {125 = {5, {125 is a perfect cube.
8
3}
• Since Ï 600 ≈ 8.434…, 600 is not a perfect cube.
Perfect cubes can be determined by cubing an integer
(e.g., ({8)3 = {512, so {512 is a perfect cube).
Perfect Square
A number for which the square root is an integer.
For example: }
49 = 7, 49 is a perfect square.
• Since Ï}
• Since Ï}
3 ≈ 1.732…, 3 is not a perfect square.
• Since Ï 0.25 = 0.5, 0.25 is not a perfect square.
8
Perfect squares can be determined by squaring whole numbers
(e.g., 152 = 225, so 225 is a perfect square).
Perimeter
The distance around a closed 2-dimensional figure or shape. In the case of a
circle, the distance around is the circumference.
Permutation
An ordered arrangement or set of elements.
3
In contrast to a combination of elements (in which order makes no difference),
changing the order changes the permutation of elements.
For example: even though they contain the same elements, the arrangements
{2, 4, 6} and {4, 2, 6} are two unique permutations.
6*
An example of three unique permutations of the same elements is
{apple, cat, car}, {car, cat, apple}, and {cat, car, apple}.
Page 39
Term
Definition
Perpendicular
Two geometric figures (e.g., lines, segments, rays) that intersect to form at
least one right angle.
June 2014
Grade
C
A
B
4
D
Perpendicular Lines
Pictograph
Perpendicular Segment and Ray
A chart that uses pictures or drawings to represent quantities.
In the example shown below, pictures of lemons are used to represent the
number of gallons of lemonade served.
Lemonade Served at the Carnival
Day
Number of Gallons
Monday
3
Tuesday
Wednesday
Thursday
Key:
= 1 gallon
Pictograph
Place Value
The value of the place a digit occupies in a number. The place value is
independent of the value of the digit occupying the place. For example, in
the decimal number 748.56, the digit 7 occupies the hundreds place (i.e., the
place value of the third place left of the decimal point is 102 or 100).
3
Page 40
June 2014
Term
Definition
Grade
Plane
A set of points that forms a flat surface that extends infinitely in all directions.
It has length and width but no height.
Informal examples that may aid students in conceptualizing a plane:
• An infinitely thin sheet of glass that extends infinitely far in all directions
• The surface of an infinitely long and wide tabletop—not the tabletop
itself, only the infinitely thin surface of the tabletop.
Point
A figure with no dimensions—it has no length, width, or height. A point is
generally indicated with a single dot and is labeled with a single capital letter
(e.g., point P). When the point appears at the end of a figure (e.g., a line
segment or a ray), it is referred to as an endpoint.
3
4
See also Ordered Pair and Vertex.
Polygon
A bounded (enclosed) two-dimensional figure. Each side of the figure is a line
segment. Each side intersects exactly two other sides at endpoints. Each
point of intersection is the intersection of exactly two sides. A polygon is
identified by the labels of its consecutive vertices.
B
C
3
A
D
E
Polygons
Polygon ABCDE
Positive Number
Any number greater than 0.
Prime Number
A whole number greater than 1 with exactly two factors, 1 and the number
itself.
For example:
• Since 7 has only two factors (1 and 7), 7 is a prime number.
• Since 9 has more than two factors (1, 3, and 9), 9 is not a prime number.
6
4
There are infinitely many prime numbers.
Page 41
June 2014
Term
Definition
Grade
Prism
A three-dimensional (solid) figure that has two congruent and parallel faces
that are polygons called bases. The remaining faces, called lateral faces, are
parallelograms (often rectangles).
Prisms are named by the shape of their bases.
5
Rectangular Prism
Triangular Prism
(On the PSSA, it may be assumed all prisms are right prisms unless otherwise
specified.)
Product
The result when one number is multiplied by one or more numbers (i.e., the
answer to a multiplication computation).
Proportion
An equation showing the equality of two ratios.
6
3= x
For example: __
}
4 16
Proportional
Relationship
3
Relationships between two variable quantities in which their ratio remains
equivalent.
For example:
• Rate of travel relationships in which the ratios of distance to time
may be written differently but remain equivalent
miles 220 miles
} = })
( e.g., 550
10 hours
4 hours
6
• Price relationships in which the ratios of cost to quantity purchased
may be written differently but remain equivalent
$13.50
$8.10
e.g., } = }
5 gallons 3 gallons
(
)
Page 42
June 2014
Term
Definition
Grade
Pyramid
A three-dimensional (solid) figure with a polygonal base and with triangular
faces that have a common vertex.
Pyramids are named by the shape of their bases.
7
Square Pyramid
Triangular Pyramid
(On the PSSA, it may be assumed all pyramids are right pyramids unless
Pythagorean
Theorem
A formula that relates the lengths of the two legs and the hypotenuse of any
right triangle. The Pythagorean theorem states the following: If a triangle is a
right triangle and has two legs with lengths a and b and a hypotenuse with
length c, then a2 + b2 = c2.
The converse is also true: If a triangle has sides with lengths a, b, and c, such
that a2 + b2 = c2, then the triangle is a right triangle. This statement is often
referred to as the converse of the Pythagorean theorem.
8
c
a
b
a2 + b2 = c2
Page 43
June 2014
Term
Definition
Grade
Quadrant
One of the four regions into which the perpendicular axes divide a coordinate
grid.
Beginning with the region in which all ordered pairs have only positive
coordinate values (the top-right region) and progressing counterclockwise
about the origin, the quadrants are named quadrant I, quadrant II, quadrant III,
and quadrant IV (note the use of Roman numerals).
y
quadrant II
quadrant I
5
x
quadrant III
quadrant IV
Quadrants
A point lies in a quadrant only if the ordered pair contains non-zero
coordinates. If either coordinate of the ordered pair is zero, then the point lies
on an axis and not in a quadrant.
Quadrilateral
3
Quadrilaterals
Page 44
June 2014
Term
Definition
Grade
Quartile
One of three values that divides a set of ordered data into four equal parts.
• first quartile (Q1)—the median of all data points less than the median of
the entire data set
• second quartile (Q2)—the median of the entire data set (Second quartile
and median are equivalent and interchangeable; however, median is
used more frequently.)
• third quartile (Q3)—the median of all data points greater than the median
of the entire data set
6*
For example, for the data set {2, 5, 7, 12, 17, 22, 23}:
• first quartile value: 5
• second quartile (median) value: 12
• third quartile value: 22
See also Median.
Quotient
The result when one number is divided by another number (i.e., the answer to
a division computation).
Radius
A line segment with one endpoint at the center of a circle and one endpoint
on the circle. The length of the radius is equal to one-half the length of the
diameter.
3
radius
5*
In common usage, the radius occasionally refers not only to the line segment,
but also to the length of the line segment that constitutes the radius.
(On the PSSA, it may be assumed that radius is the line segment, not the
measurement of the line segment unless otherwise specified. If there is a
context in which radius is intended to imply a measurement, the context must
clearly, absolutely, and indisputably make that assertion.)
See also Diameter.
Range (of Data)
The difference between the greatest and the least values in a set of data.
For example:
• For the data set {1, 7, 9, 11}, the range is 11 – 1 = 10.
• For the data set {5, 7, 12, 23, 29}, the range is 29 – 5 = 24.
6
Page 45
June 2014
Term
Definition
Grade
Rate
A ratio that compares two quantities with different measurements (e.g.,
distance compared to time; height, in inches, compared to width, in inches).
Rate is a measure of change.
For example:
• miles per hour
• dollars : pounds
• change in y compared to change in x (i.e., slope)
6
See also Ratio.
Ratio
A comparison of two numbers, quantities, or expressions by division. It is
often written as a fraction, but not always (e.g., 2
}, 2:3, 2 to 3, and 2 ÷ 3 all
3
represent the same ratio).
Rational Number
6
Any number that is equivalent to a fraction written as an integer over a
counting number. The set of rational numbers includes all of the integers since
each integer can be written as that number over one.
For example:
4, since it is a fraction of an integer over a counting number
• }
7
• 27, since it is equivalent to 27
}
1
6
5, since it is equivalent to { 29
}
• {3 }
8
8
• 3.71, since it is equivalent to 371
}
100
}
• 24.3, since it is equivalent to 24 1
}
3
94,619
}
• 0.94713, since it is equivalent to }
99,900
See also Irrational Number.
Ray
A part of a line that has one endpoint and continues infinitely in one direction
or on one side of that point. A ray is identified by two points: first its endpoint
and then another unique point on the ray.
A
B
4
Ray AB (####$
AB)
Page 46
June 2014
Term
Definition
Grade
Rectangle
A parallelogram with all angles congruent. Each of the angles in a rectangle
is 90°.
K*
Rectangle
Rectangular Prism
A three-dimensional (solid) figure which has exactly six faces. All six faces are
rectangles.
5
Rectangular Prism
Reflection
The transformation of a figure that produces the mirror image of the original
figure. As a result of the transformation, the line over which the reflection
occurs becomes a line of symmetry. Because the reflected image is congruent
to the original image, a reflection is referred to as a rigid transformation.
Informally, a reflection can be thought of as a “flip” of the original figure.
8
Reflection
See also Line of Symmetry.
Regular Polygon
A polygon in which all sides are congruent and all angles are congruent.
4*
Regular Polygons
Two special types of regular polygons are equilateral triangles and squares.
Page 47
Term
Definition
Relation
Any set of ordered pairs.
June 2014
Grade
For example:
• {(5, 9), ({3, 7), (15, 2), ({3, 9)}
• {(insect, ant), (reptile, lizard), (bird, goose), (mammal, deer)}
• {(x + 2, 3 × w), (a, b), (x + a, w – b), (a, 3 × w)}
y
5
4
3
2
1
•
Repeating Decimal
Number
–5 –4 –3 –2 –1
–1
–2
–3
–4
–5
8
1 2 3 4 5
x
A decimal number in which the fractional part (the part to the right of the
decimal point) is non-terminating and extends infinitely in a repeating
sequence of digits. When written, a bar may be written above the repeated
}
digits (e.g., 0.333… may be written as 0.3). When a repeating decimal is
written in decimal notation without the bar, an ellipsis (…) must be used to
indicate the decimal does not terminate; also, three repetitions of the repeated
digit(s) and/or some indication of which digits are repeated must be included.
Only those numbers written under the bar are repeated infinitely. All repeating
decimal numbers are rational numbers.
8
For example:
}
• 24.3 = 24.333… (the 3 repeats infinitely)
}
• 0.94713 = 0.94713713713… (the 713 repeats infinitely)
}
• 193.40 = 193.404040… (the 40 repeats infinitely; note the 0 cannot be
ignored)
See also Terminating Decimal Number.
Page 48
Term
Definition
Rhombus
A parallelogram with all sides congruent. The plural of rhombus is rhombi.
June 2014
Grade
3
Rhombus
Right Angle
An angle that measures exactly 90º. A right angle may be marked with a small
square in the interior of the angle.
4
Right Angle
Right Triangle
A triangle in which an interior angle is a right angle.
4
Right Triangle
See also Acute Triangle and Obtuse Triangle.
Rotation
The transformation of a figure that moves the figure by rotating it about a
fixed point. Often the point about which the original figure is rotated and the
degrees of rotation are stated (e.g., a 90° clockwise rotation about point A).
Because the rotated image is congruent to the original image, a rotation is
referred to as a rigid transformation. Informally, a rotation can be thought of as
a “turn” of the original figure.
8
Rotation
Page 49
June 2014
Term
Definition
Grade
Scale Drawing
A drawing that is geometrically similar to an original figure or object. In a
scale drawing, the linear measurements may change but the proportional
relationships of those measurements are preserved (i.e., length measurements
in the scale drawing remain uniformly proportional to length measurements
in the original figure). The angle measurements in a scale drawing and the
original object or figure are congruent.
7
See also Proportional Relationship.
Scale Factor
The number by which the length(s) of a geometric object is multiplied to
generate a similar geometric object. The scale factor is the magnitude of a
dilation.
If a scale factor is greater than one, the dilated figure is larger than the original
figure. If the scale factor is less than one, the dilated figure is smaller than the
original figure. If the scale factor is one, the dilated figure is congruent to the
original figure (i.e., the figure does not change).
7*
In some cases, the scale factor is a negative number. A negative scale factor
results in both a dilation and a reflection. (Negative scale factors are generally
only used when the original figure appears on a coordinate grid.)
Scalene Triangle
A triangle in which no two sides are congruent (i.e., all three sides have
different lengths).
4*
Scalene Triangle
See also Equilateral Triangle and Isosceles Triangle.
Page 50
June 2014
Term
Definition
Grade
Scatter Plot
A plot that represents discrete bivariate data. The data points are represented
by ordered pairs marked on a coordinate grid.
In addition to visually representing data, scatter plots often serve as the
geometric basis for derivation and application of lines of best fit.
Time Needed to Paint Houses
in a Neighborhood
Number of Hours
y
80
8
60
40
20
x
0
1
2
3
4
5
6
7
Number of Painters
8
9
Scatter Plot
See also Line of Best Fit.
Scientific Notation
A form of exponential notation created by writing a number as the product of a
decimal number multiplied by a power of 10 (e.g., 103). If the original number is
positive, the decimal number must be greater than or equal to 1, but less than
10. If the original number is negative, the decimal number must be less than or
equal to {1, but greater than {10.
A number is written in scientific notation by “floating” the decimal point in the
original number to a position where it is preceded by a single, nonzero digit
and then multiplying that number by the greatest power of ten less than or
equal to the original number.
8
For example:
• The scientific notation of 23,911.1862 is 2.39111862 × 104.
• The scientific notation of 0.00531 is 5.31 × 10{3.
Scientific notation is generally used to represent numbers that have either very
large or very small absolute values.
Page 51
June 2014
Term
Definition
Grade
Similar
Geometric figures in which the measures of corresponding sides are uniformly
proportional and the measure of corresponding angles are congruent. In
similar figures, the linear measurements may be different but the proportional
relationships of those measurements are preserved (i.e., length measurements
in the one figure remain uniformly proportional to length measurements in the
other figure). Similar figures are dilations of each other.
An informal definition of similar figures is figures with the same shape but not
necessarily the same size.
8
Y
B
A
X
C
Z
Similar Triangles
Similar Quadrilaterals
Figures that are congruent are also similar.
Slope
The ratio of the vertical change compared to the horizontal change
between two points on a coordinate grid. Slope is often expressed as
rise or change in y. A vertical line has an undefined slope. A horizontal line
}
}}
run
change in x
has a slope of 0. Note that slope is a rate.
y
(x2, y2)
(x1, y1)
8
rise
run
x
Slope
y 2 – y1
The variable m is often used to represent slope (e.g., m = }
x2 – x1 ,
y = mx + b).
Page 52
Term
Definition
Sphere
A three-dimensional (solid) figure in which all points on the surface are the
same distance from the center.
June 2014
Grade
8
Sphere
Square
A parallelogram with all sides congruent and all angles congruent. Thus, a
square is also a rectangle and a rhombus.
K*
Square
Square Root
One of two equal factors (roots) of a number or expression. Informally, it can
be thought of as “the number, when multiplied by itself, has a product equal to
a given number.”
Note that any positive number has two square roots: one positive and one
negative. The unique nonnegative square root of a nonnegative number is the
principal square root. The square roots
of 25 are 5 and –5; the principal square
}
root of 25 is 5 and can be written Ï25 = 5.
8
For example:
}
• Ï 9 = 3 since 3 × 3 = 9 and 3 is nonnegative
}
• Ï 0.36 = 0.6 since 0.6 × 0.6 = 0.36 and 0.6 is nonnegative
}
• Ï 49w4 = 7w2 since 7w2 × 7w2 = 49w4 and 7w2 is nonnegative
Page 53
June 2014
Term
Definition
Grade
Stem-and-Leaf
Plot
A plot that represents discrete numerical data. In the display, a bar separates
common digits in larger place values from the smaller digits.
The numbers to the left of the bar are the stems and the numbers to the right
of the bar are the leaves. Generally, the leaves are the digits in the ones place
of all the numbers in a data set and the stems are the common digits in the
place values greater than the ones place.
Number of Sit-Ups
Each tens digit
is called
a stem.
3
4
5
4
0
0
6
3
0
8
6
1
8
7
2
7
Each ones digit
is called
a leaf.
6*
Key
3 | 6 = 36
Stem-and-Leaf Plot
Straight Angle
An angle with a measure of exactly 180°. A straight angle created by two rays
forms a line.
Subtrahend
An expression that is subtracted from another expression.
For example:
• In the computation 29 – 11 = 18, 11 is the subtrahend.
• In the expression (3 + x) – 7w, 7w is the subtrahend.
Sum
The result when adding two or more numbers (i.e., the answer to an addition
computation).
Supplementary
Angles
Two angles for which the sum of their measures is 180°.
5*
4
3
If two supplementary angles are also adjacent angles, they form a straight
angle.
Each of two supplementary angles is referred to as the supplement of the
other angle (e.g., a 125° angle is the supplement of a 55° angle).
7
55˚
125˚
Supplementary Angles
See also Complementary Angles.
Page 54
June 2014
Term
Definition
Grade
Surface Area
The sum of the areas of all the faces of a three-dimensional (solid) figure or
object.
Tally Chart
A table or chart in which tally marks (in contrast to numbers or pictures) are
used to record data.
6
Favorite Stores
Type of
Store
Number of
Students
3
pet
book
game
hardware
Tally Chart
Terminating
Decimal Number
A decimal number that can be written, in its entirety, with a finite number of
digits.
8
See also Repeating Decimal Number.
Theoretical
Probability
A likelihood of an outcome based on the number of expected favorable
outcomes compared to the number of possible outcomes. A theoretical
probability is determined prior to any trials.
The value of a theoretical probability (P) is determined by the following
formula:
theoretical number of favorable outcomes
P (favorable outcome) = }}}}
theoretical number of possible outcomes
For example:
• Probability of flipping heads in one trial is 1
}
2
s 1 is the theoretical number of favorable outcomes (heads)
s 2 is the theoretical number of possible outcomes (heads or tails)
8*
2
• Probability of the first snow occurring on a Tuesday or a Wednesday is }
7
s 2 is the theoretical number of favorable outcomes (Tuesday or
Wednesday)
s 7 is the theoretical number of possible outcomes (7 days in a week)
See also Experimental Probability.
Page 55
June 2014
Term
Definition
Grade
Time (analog)
Time displayed by an analog clock. Analog clocks display continuous time.
Traditional two- or three-hand clocks are examples of clocks that display
analog time.
11
12
1
10
2
9
3
3
8
4
7
6
5
Analog Clock
Time (digital)
Time displayed as digits, as seen on digital clocks. Digital time shows each
unit of time separated by colons. Digital clocks typically display only wholenumber hours, minutes, and/or seconds. Digital times may refer to either
elapsed time or the time of the day.
For example:
• 2:57 represents 2 hours, 57 minutes
• 11:03:20 represents 11 hours, 3 minutes, 20 seconds
• 7:45 P.M. represents 7 hours, 45 minutes after noon and is read as
“seven forty-five P.M.”
3*
(On the PSSA, it may be assumed all digital times begin with the hour unless
Transformation
The application of a rule that may change the size or location of a geometric
figure. Application of the rule is termed a “mapping.” Transformations may
include translation, reflection, rotation, or dilation.
A rigid transformation is one in which the new figure is congruent to the
original figure. A non-rigid transformation is one in which the new figure is not
congruent to the original figure (the new figure may be similar to the original
figure, although this is not always the case).
8
Page 56
June 2014
Term
Definition
Grade
Translation
The movement of a figure to a new position without any dilation, rotation, or
reflection. It is a transformation in which the size and orientation of the original
figure remain constant but the location in a plane changes. Because the
translated image is congruent to the original image, a translation is referred
to as a rigid transformation. Informally, a translation can be thought of as a
“slide” of the original figure.
8
Translation
Transversal
A line that intersects two or more other lines. The lines intersected by a
transversal may or may not be parallel.
The relationships of angles formed by the intersection of two lines and a
transversal are frequently encountered in the study of geometry.
f
2
1
4
5
8
3
l
7
6
7
m
Line f is a transversal through parallel lines l and m.
See also Alternate Exterior Angles, Alternate Interior Angles, and
Corresponding Angles.
Trapezoid
A quadrilateral with exactly one pair of parallel sides.
6
Trapezoids
Page 57
June 2014
Term
Definition
Grade
Triangle
A polygon with exactly 3 sides. A triangle may be classified by its side lengths
(i.e., equilateral triangle, isosceles triangle, or scalene triangle), or by its angle
measures (i.e., acute triangle, obtuse triangle, right triangle, or equiangular
triangle).
K*
Triangles
Triangle Inequality
Theorem
The theorem that asserts the sum of the lengths of any two sides of a triangle
is greater than the length of the third side.
c
b
7
a
a+b>c
Two-Way Table
a+c>b
b+c>a
A table that shows the relationship between two sets of categorical variables.
The entries in the table are either frequency counts (numerical values) or
relative frequencies (ratios or percents).
High Temperatures during the Month
January
February
Above 40°F
14
18
40°F or Colder
17
10
8
High Temperatures during the Month
January
February
Above 40°F
45%
64%
40°F or Colder
55%
36%
Two-Way Tables
Unit Price
The price of a single item or unit (e.g., $3.50 per pound).
4*
Page 58
Term
Definition
Unit Rate
The ratio of a quantity to a single unit of comparison.
June 2014
Grade
For example:
52 miles
• 52 miles per hour – or – 52 miles : 1 hour – or – }
1 hour
8.3 pounds
• 8.3 pounds per gallon – or – 8.3 pounds : 1 gallon – or – }}
1 gallon
6
4 beats
• 4 beats per measure – or – 4 beats : 1 measure – or – }
1 measure
$2.98
• $2.98 per pound – or – $2.98 : 1 pound – or – }
1 pound
See also Constant of Proportionality and Unit Price.
Unit Square
A square with each side 1 unit in length. The area of a unit square is
1 square unit.
1 unit
area = 1 square unit
1 unit
3
Unit Square
Variable
A letter or symbol that represents a missing or unknown value. Generally, the
letter is lowercase and italicized.
For example:
• In the expression 5w + 17, the variable is the w.
• In the equation 3 + = 9, the variable is the .
• In the formula y = mx + b, the variables are the y, m, x, and b.
6
Note: not all special characters are variables. For example, the Greek letter
π (pi) represents a specific value (3.14159265…).
Page 59
June 2014
Term
Definition
Grade
Venn Diagram
A diagram that represents the relationship between sets of data (either
numerical or categorical). The diagram typically consists of data entered into
two or more circles—distinct or intersecting—drawn inside a rectangle. The
rectangle represents the universal set and the circles represent subsets. Data
that are in two or more of the subsets will appear in the intersection of the
circles representing those subsets.
In the Venn diagram below, the left circle contains the prime numbers less
than 20 (2, 3, 5, 7, 11, 13, 17, and 19) and the right circle contains the odd
whole numbers less than 20 (1, 3, 5, 7, 9, 11, 13, 15, 17, and 19). Since the
numbers 3, 5, 7, 11, 13, 17, and 19 are both prime and odd, they appear in
the intersection (overlap) of the two circles; outside of the circles are the even,
nonprime whole numbers less than 20 (0, 4, 6, 8, 10, 12, 14, 16, 18).
Whole Numbers Less than 20
4*
0
10
Odd
Prime
4
3
2
6
5
7 11 13
17 19
12
1
14
9
15
16
8
18
Venn Diagram
This representation of data is named after the English mathematician/logician
John Venn (1834–1923).
Page 60
June 2014
Term
Definition
Grade
Vertex
A point where lines, rays, line segments, two sides of a two-dimensional
(plane) figure, or three edges of a three-dimensional (solid) figure meet. A
vertex is the single point that geometric figures have in common when they
intersect. The plural of vertex is vertices.
For example:
• The vertex of an angle is the point at which the rays that form the angle
intersect.
• A vertex of a pyramid is a point at which three faces intersect.
• A vertex of a square is one of the “corners” (a point at which two sides
intersect).
• The vertex of a cone is the point opposite the base.
4
vertex
vertex
angle
cone
Vertex
Vertical Angles
The pair of angles with the same vertex on opposite sides of two intersecting
lines. Vertical angles are congruent.
M
N
20˚
R
20˚
P
7
Q
/MRP and /NRQ are vertical angles.
Note: vertical angles are not necessarily oriented vertically.
Volume
The amount of space (in cubic units) that a three-dimensional (solid) figure
occupies or contains. Units such as cubic meters (m3), cubic inches (cu in.),
gallons (g), liters (L), and fluid ounces (fl oz.) are used to measure volume.
3
A counting number or zero. Any number from the set of numbers represented
by {0, 1, 2, 3, …}. A whole number is sometimes referred to as a non-negative
integer.
3
x-Axis
The horizontal axis of a coordinate grid.
5
y-Axis
The vertical axis of a coordinate grid.
5
Whole Number
Page 61
Assessment Anchors and Eligible Content
with Glossary
June 2014
Copyright © 2014 by the Pennsylvania Department of Education. The materials contained in this publication may be
duplicated by Pennsylvania educators for local classroom use. This permission does not extend to the duplication
of materials for commercial use.
Cover photo © Hill Street Studios/Harmik Nazarian/Blend Images/Corbis.

PSSA Mathematics Glossary

Transcription

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